Nuprl Lemma : fpf-compatible-singles
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[x,y:A]. ∀[v:B[x]]. ∀[u:B[y]].
x : v || y : u supposing (x = y ∈ A) ⇒ (v = u ∈ B[x])
Proof
Definitions occuring in Statement :
fpf-single: x : v,
fpf-compatible: f || g,
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
fpf-compatible: f || g,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
prop: ℙ,
top: Top,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
fpf-single: x : v,
fpf-dom: x ∈ dom(f),
pi1: fst(t),
deq: EqDecider(T),
or: P ∨ Q,
false: False,
guard: {T},
eqof: eqof(d),
iff: P ⇐⇒ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
rev_implies: P ⇐ Q
Lemmas referenced :
deq_wf,
subtype_rel_wf,
subtype_rel_self,
equal_wf,
top_wf,
fpf-single_wf,
fpf-dom_wf,
assert_wf,
fpf_ap_pair_lemma,
deq_member_cons_lemma,
deq_member_nil_lemma,
istype-assert,
bor_wf,
bfalse_wf,
eqof_wf,
false_wf,
istype-void,
subtype_rel-equal,
equal_functionality_wrt_subtype_rel2,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
or_functionality_wrt_uiff2,
safe-assert-deq
Rules used in proof :
equalityTransitivity,
axiomEquality,
dependent_functionElimination,
isect_memberFormation,
universeEquality,
because_Cache,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
functionExtensionality,
applyEquality,
functionEquality,
voidEquality,
voidElimination,
isect_memberEquality,
hypothesis,
lambdaEquality,
sqequalRule,
instantiate,
hypothesisEquality,
cumulativity,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
productEquality,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
Error :memTop,
lambdaFormation_alt,
productElimination,
productIsType,
setElimination,
rename,
universeIsType,
unionEquality,
unionIsType,
equalityIstype,
inhabitedIsType,
unionElimination,
independent_functionElimination,
independent_isectElimination,
dependent_set_memberEquality_alt,
independent_pairFormation
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[x,y:A]. \mforall{}[v:B[x]]. \mforall{}[u:B[y]].
x : v || y : u supposing (x = y) {}\mRightarrow{} (v = u)
Date html generated:
2020_05_20-AM-09_02_57
Last ObjectModification:
2020_01_26-PM-00_00_44
Theory : finite!partial!functions
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